They will visit the winners of Arsenal and Liverpool in the last eight which does not make their task any easier. Home debutant Lacina Traore made a bright start to his Everton career when he backheeled the home side ahead on four minutes only for Jonathan de Guzmán to nod the visiting side level 11 minutes later.
Roberto Martinez's side were much more compact in the second period and regained the lead when Traore's replacement Naismith slid the ball under Gerard Tremmel after Neil Taylor was short with a passback to his goalkeeper.
Leigton Baines sent Tremmel the wrong way from the penalty spot after Naismith was felled by Jazz Richards as Everton moved 3-1 clear on 72 minutes.
Naismith did not finish the match as he was taken off with concussion on 86 minutes having collided with Jordin Amat moments earlier, but his side were already into the last eight.
They were comfortable winners on the day with Martinez bidding to win the FA Cup for a second straight year after his success managing Wigan last season.
4' - DREAM START FOR TRAORE: Traore backheels a loose ball into the net from a Distin pass. Goes in off Richards and Everton lead 1-0. What a start to life at Everton.
16' - DE GUZMAN LEVELS: De Guzman pops up to bullet a header into the net from the Taylor cross. Timed that to perfection as he arrived on the run to level at 1-1. No chance for Joel. Game back in the melting pot.
21' - PIENAAR HITS THE BAR: Pienaar clips an effort from the Baines cross that comes off the bar. Breaks to Traore, but his header is held by Tremmel. So close to a second goal for Everton.
65' - NAISMITH NUDGES EVERTON AHEAD: Naismith slides the ball under Tremmel, and Everton lead 2-1. Awful pass by Taylor trying to knock the ball back to his goalkeeper. Short of pace, and he is punished by Naismith.
73' - BAINES SEALS THE WIN FOR THE HOME SIDE: Baines slides the ball his right. Tremmel dives to his right. And that is a 3-1 lead. Surely game over now.
Steven Naismith has scored four goals in his last five appearances for Everton, including three in his last two FA Cup matches.
Leighton Baines' converted spotkick was his first goal at Goodison Park since February 2013.
Since leaving Swansea in 2009, Roberto Martinez has now won three of his seven meetings in all competitions with the club.
MAN OF THE MATCH
Sylvain Distin (Everton): A composed performance at the back. A real rock at the back with Everton showing their true class in the second period.
TALKING POINT: Can Everton win their first major trophy since lifting the FA Cup in 1995? - Everton look to be a decent shout to win the trophy. They might not have enough to finish in the top four of the Premier League, but certainly have enough about them to upset any of England's top four on any given day. They will need to be having landed Arsenal or Liverpool in the last eight.
Everton: Robles 6, Coleman 8, Baines 7, Jagielka 7, Distin 8, McCarthy 7, Barry 8, Mirallas 7, Pienaar 7, Barkley 6, Traore 7..subs: Deulofeu 6, Naismith 8, Osman 7
Swansea City: Tremmel 7, Richards 5, Bartley 6, Amat 6, Taylor 5, Canas 5, De Guzman 7, Hernandez 6, Routledge 6, Lamah 5, Vazquez 5..subs: Williams 6, Dyer 6, Lita 6
THE MATCH IN A TWEET